\documentclass[12pt,a4paper]{report}
\usepackage{graphicx}
\newtheorem{procedure}{Procedure}
\title{Blind Subband Adaptive Equalization}
\author{J\"orgen Nordberg}
\begin{document}
\maketitle
\begin{abstract}
It is predicted that a large portion of future wireless
communication capacity will be used to provide wireless data
services. This new emphasis results in a significant change in
substantial parts of the wireless infrastructure, it is important
to have low bit error transmission links to get Quality of Service
similar to the internet. Future services will also demand higher
data rates. These factors mean that there will be a significant
need for good quality equalization schemes in the receiver to
reduce the different effects of the radio channels, such as
intersymbol interference and multipath propagation. The use of
linear equalizers leads to high numerical complexity filters even
for short delay spread channels. High complexity equalizers
implies slow convergence and high numerical load per processed
symbol. To save the amount of training or pilot signals and thus
increasing the useful data bits, a blind adaptation of the
equalizer weights is commonly used. A blind adaptation algorithm
in combination with a long equalizer result in even slower
convergence. It is therefore desirable to improve the convergence
properties of such equalizer schemes. A blind delayless subband
equalizer structure is shown to improve the convergence properties
of the equalizer. This equalizer consists of a combination of a
fullband filtering and and a subband adaptation.
In this paper a novel filter bank design and also improved
subband--to--fullband filter transformation methods is presented.
By optimizing the filter bank design, and improving the
subband--to--fullband transformation, considerable improvements
have been achieved both when it comes to convergence speed and the
level of equalization obtained.
Simulation results show that the new subband equalization
structure has a 12 times faster convergence than its fullband
counterpart. The bit error rate is almost the same in both
schemes. Thus, there are only advantages by using the suggested
equalizer implementation compared to a conventional fullband
implementation. The convergence improvement combined with
computational savings make it a very attractive technique to use.
\end{abstract}
\chapter{Introduction}
Wireless data services can be provided either as a part of the
mobile phone network service or from a wireless local area network
(WLAN). WLAN's have become popular during the last couple years
and there is services available in airports and other public
areas. It is also common for offices and homes.
WLAN standards like the institute of electrical and electronics
engineers (IEEE) 802.11a, and Hiperlan II supports data traffic
up to 52 Mbit/s in the physical layer, or 32 Mbit/s in the data
layer. Both systems work in the 5 GHz band. Hiperlan II is defined
by European telecommunication standardization institute (ETSI),
while the IEEE 802.11a standard is developed by IEEE.
The WLAN systems support high speed data traffic over rather
short distances. The transmitted electromagnetic signal is
attenuated and encounter multipath propagation before reaching the
receiving antenna. Depending on the data-rate flat fading or
frequency selective fading will occur. Since, the carrier
frequency lies in the 5 GHz band, the transmitted wave will not
easily penetrate obstacles and thus the receiver will rarely have
any line of sight (LOS) signal. Consequently, the received signal
will mainly consist of reflected signals. This fact in combination
with the high data rate, means that the channel encounter severe
frequency selective fading. To recover the transmitted signal from
the effects of fading, an equalization scheme must be employed at
the receiver. The equalization is usually made adaptive to be
able to track channel variations.
Both Hiperlan II and IEEE 802.11a use orthogonal
frequency-division multiplexing (OFDM). OFDM is a multi-carrier
(MC) modulation where a data stream is divided and transmitted
into several low-rate orthogonal frequency bands. However, it is
stated in \cite{falconer} that the same receiver complexity can be
achieved by using a signal carrier (SC) modulation approach and a
frequency domain equalizer. The advantages of using SC modulation
are that it is a more well-proven technology and has a lower
peak-to-average power ratio in the transmitted signal than the MC
modulation system, there will be less stringent linearity
requirements of the RF part of the system. Therefore, this paper
will study the equalization problem for SC-modulation and use a
subband implementation to improve the convergence rate.
Equalizers are usually using a pre-known data sequence
(pilot/training sequence) in each data block to estimate the
equalizer filter weights, but there is also a variety of blind
equalizers. Blind equalizers use properties in the data format to
recover the transmitted symbol. Non-blind equalizers are data
aided and use pilot sequences. The pilot signal occupy space
where ordinary data bits could have been transmitted, a reduction
of the system capacity will therefore occur. The length and the
separation distance of the pilot sequence is restricted to avoid
unacceptable capacity loss, and therefore the equalizer control
algorithm needs to be able to track channel variations between
each pilot sequence. When the channel variations between two pilot
sequences are moderate, a decision-directed (DD) or
decision-feedback (DFE) scheme can be used on the non-pilot bits
to track the channel variations and enhance the performance of the
equalizer. This works well only in slowly changing environments,
such as in an office environment and radio links. If the channel
variations are too rapid between each training sequence the DD and
the DFE equalizers will not perform well.
To avoid problems with short, sparsely placed pilot sequences
and a fast changing channel, equalizers can use pre-known signal
characteristics to estimate equalizer coefficients. This type of
equalizers, which use a priori information instead of pilot
sequences, are usually called blind-equalizers. Since, the signal
characteristics of the transmitted signal are used, all
transmitted bits can be used to estimate equalizer parameters.
Blind algorithms are characterized by slow convergence and are
also subject to phase ambiguity problems. The slow convergence is
not a large problem since the blind equalizer can utilize all
received data to estimate the equalizer weights. The phase
ambiguity can be solved by combining the equalizer with a phase
recovering scheme.
Linear channel equalization is a weighted inverse filtering problem
and even a short delay spread can result in a computationally
complex equalizer with hundreds of filter coefficients to estimate.
Recently, subband techniques have been considered as means to
reduce both the computational complexity and improve the
convergence speed of the equalizer
\cite{Stewart, Weiss}.
In this paper, a closed--loop delayless subband constant modulus
algorithm (CMA) equalizer will be presented and compared with its
fullband counterpart. The subband equalization scheme consists of
two parallel main parts: one fullband equalization filter part that
continuously equalize the received signal and one subband part where
the fullband equalization filter weights are estimated
\cite{MThi95}. The blind Godard algorithm \cite{godard}, also
known as CMA, is used to estimate the subband equalizer
coefficients. In order for the CMA to work, the signal
must fulfill the constant modulus (CM) criterion. Quadrature
phase shift keying (QPSK) modulation is used in this study.
Since, its signal constellation fulfills the CM criteria.
However, the CM criteria cannot be guaranteed in the different
subbands, therefore is the CMA cost function applied to the
fullband signal, and an CM error is created.
The fullband CM error is then fed back into the
subbands. The estimated equalization coefficients in each subband
are transformed into a fullband equivalent filter. Two different
transformation methods are used and compared. The performance of
the subband adaptive equalizer depends on the filter bank and the
subband--to--fullband transformation. A filter bank design method
which emphasizes low aliasing in the subband signals is presented.
This new optimal design method improves the subband performance
considerable. Average amplitude fluctuation and bit error rate
(BER) are used as the performance measures to evaluate the closed
loop delayless subband CMA equalizer. Simulation parameters which
have been studied and varied are filter bank design methods,
signal to noise ratio (SNR), the number of subbands, fullband to
fullband transformation methods and equalizer lengths.
The report is organized as follows: the system model is introduced
in Chapter \ref{sec2}. Chapter \ref{ch:clscma} presents the
closed--loop subband CMA equalizer. The blind subband equalizer is
evaluated through simulations. The results from these simulations
are presented in Chapter \ref{ch:res}. Finally, conclusions are
drawn in Chapter \ref{ch:con}.
\chapter{System Model}\label{sec2}
Consider a communication system given in block diagram form in
Figure \ref{channel}. The transmitted bit $b(n)$, where $n$
denotes the discrete time index, is independently generated with
equal probabilities and modulated using QPSK. In QPSK four
different signal alternatives are used
\begin{equation}
s_{i}(t)= \left \{
\begin{array}{ll}
\sqrt{\frac{2E}{T_s}}\cos \left[ 2\pi f_{c}(2i-1) \frac{\pi}{4} \right ] & \qquad 0 \leq t \leq T_{s} \\
& \\
0 &
\end{array}
, \right.
\end{equation}
where $i=1,2,3,4$, E is the transmitted signal energy per symbol,
$T_{s}$ is the symbol duration and $f_c$ is the carrier frequency.
The transmitted bits are mapped to the four signal alternatives
by Gray coding, see Table \ref{table:gray}.
\begin{table}
\centerline{
\begin{tabular}{|c|c|}
\hline
bit pattern & Phase Offset\\ \hline
10 & $\pi / 4$ \\
\hline
00 & $3\pi / 4$ \\
\hline
01 & $5\pi / 4$ \\
\hline
11 & $7\pi / 4$ \\
\hline
\end{tabular}}
\caption{Gray coding table.}
\label{table:gray}
\end{table}
The modulated signal $s(t)$ is then transmitted through a channel
modelled as a time varying FIR filter with the impulse response
${\bf c}(t)=[c_{0}(t),\cdots,c_{K-1}(t)]^{T}$, where $K$ and
$[\cdot]^{T}$ denote the length of the channel and the vector
transpose, respectively.
\begin{figure}[!!!h]
%\centerline{\includegraphics[scale=.7]{channel.eps}}
\centerline{\includegraphics[scale=.7]{sysmodel.eps}}
\caption{Communication system model.} \label{channel}
\end{figure}
The received continuous time signal $x(t)$ is given by
\begin{equation}
x(t)=\sum \limits^{K}_{n=0} c_{k}(t)s(t-\tau_{k})+w(t),
\label{eq:tch}
\end{equation}
where $\tau_{k}$ denotes the delay associated with the k$^{th}$
path and $w(t)$ is additive white Gaussian noise. In this study a
Hiperlan II \cite{channel, channel1} inspired channel model is
considered, i.e. a quasi-static sparse frequency selective channel
with additive white Gaussian noise $w(t)$. Thus, (\ref{eq:tch})
will reduce to
\begin{equation}
x(t)=\sum \limits^{K}_{k=0} c_{k}s(t-\tau_{k})+w(t).
\label{eq:tch2}
\end{equation}
The channel has 100 taps of which 40 have a significant value
(non-zero value). The significant taps are randomly generated
and placed in the impulse response. The magnitude and phase
responses of a typical channel are given in Figure \ref{fig_channel}.
\begin{figure}[!!!h]
\centerline{\includegraphics[scale=.5]{ch_carr2.eps}}
\caption{Channel characteristics.}
\label{fig_channel}
\end{figure}
At the receiver $x(t)$ is sampled at symbol rate $T_s$ and the
discrete-time received signal is obtained
\begin{equation}
x(n)=x(nT_{s}),\quad 0 \leq n \leq \infty.
\end{equation}
\chapter{Closed Loop Subband CMA Equalizer}\label{ch:clscma}
The performance of non-blind equalizers is highly dependent on the
quality of the training/desired signal \cite{monzigo}. Blind
techniques avoid this disadvantage by exploiting {\em a
priori} information about signal characteristics. The blind
algorithm used in this study is the low computationally complex
Godard (CMA) algorithm \cite{godard}. This algorithm was also
independently developed and extended by Treichler et. al.
\cite{treich}. Since, CMA exploits CM properties, the Godard error
signal is calculated on the fullband signal. Subsequently, the
error signal is feed back through the filter bank to the subband
filter adaptation algorithms, i.e. closed loop implementation, see
Figure \ref{subband_eql}.
\begin{figure}[!!!!h]
\centerline{\includegraphics[scale=.9]{subeq.eps}}
\caption{A closed--loop subband CMA equalizer.}
\label{subband_eql}
\end{figure}
The received signal $x(n)$ is divided into $M$ frequency
bands by using a uniformly modulated filter bank (UMF).
The UMF is formed by several modulated versions of
a prototype filter \cite{haan,Nordberg}.
Thus, transfer function of the $m^{th}$ subband filter $H_{m}(z)$ is given by
\begin{equation}
H_{m}(z)=H(zW^{m}_{M})=\sum^{L_{h}-1}_{l=0}
h(l)(zW^{m}_{M})^{-l},
\label{eq:umf}
\end{equation}
where $W_{M}=e^{-j2\pi /M}$, $m \in [0,M-1]$, $h(l),\ 0 \leq l \leq L_{h}$
denotes a protype filter of order $L_{h}$ and M is the number of subbands. Thus the $m^{th}$
subband signal in the frequency domain, $X_m(z)$, is formed by
\begin{equation}
X_{m}(z)=X(z)H_m(z), \qquad 0 \leq m \leq M-1.
\end{equation}
Polyphase decomposition is used to efficiently implement the UMF. Due to polyphase decomposition
the decimation can be moved before the subband filtering
\cite{Vai93}. By dividing the received signal into
several frequency bands it is possible to decimate the
signal with a decimator factor $D$. By including the polyphase decomposition and the decimation,
(\ref{eq:umf}) is given by
\begin {equation}
\label{eq:hl1}
H_m(z) = \sum_{d=0}^{D-1}(W^mz)^{-d}\sum_{n=-\infty}^{\infty}h(nD+d)(W^mz)^{-nD}.
\end {equation}
In critically decimated filter banks the decimation factor is set to $D=M$ and aliasing will
decrease the performance of the subband schemes. This is, of course, more noticeable with an increasing
order of subbands. To reduce the impact of aliasing the filter bank
is oversampled with a factor of two, i.e. the subband signals are decimated by a factor
$D=\frac{M}{2}$ \cite{grbic}. Including $D=\frac{M}{2}$ in (\ref{eq:hl1}) results in the following
expression of $X_m(z)$
\begin{equation}
X_m(z)=X(z)H_m(z) =X(z) \sum_{d=0}^{\frac{M}{2}-1}(W^mz)^{-d}\sum_{n=-\infty}^{\infty}h(n
\frac{M}{2}+d)(W^mz)^{-nM/2}.
\end{equation}
where
\begin {equation}
W^{-mnM/2}=(e^{j \pi m})^n
= \left\{ \begin{array}{ll}
(-1)^n & \textrm{m odd} \\
1 & \textrm{m even} \\
\end{array}. \right.
\label{eq:W}
\end {equation}
To further reduce the effects of aliasing, a filter bank design method that minimizes
the aliasing in the subband signals will be presented in Section \ref{fbd}.
The equalizer coefficients in each frequency band are estimated by using
CMA. The impulse response for the fullband equalizer is the inverse discrete
Fourier transform (IDFT) of the fullband frequency response, which is
obtained by stacking the subband responses \cite{MThi95} together.
In Section \ref{stack} two different subband--to--fullband transformation methods are presented.
The fullband equalizer impulse response is denoted by ${\bf g}=[g(0),\cdots,g(L-1)]^{T}$,
and the filter length $L$ is chosen so that the signal $b(n)$ can be recovered. Then, the
output signal from the equalizer is given by
\begin{equation}
y(n)=\sum^{L-1}_{l=0}g(l)^{*}x(n-l).
\end{equation}
where $ ^*$ denotes the complex conjugate.
\section{Filter Bank Design} \label{fbd}
Since, an UMF is used to split the signals
into equally spaced modulated frequency bands, the
design of the filterbank is reduced to designing a prototype filter
of length $L_{h}$ and impulse response ${\bf h}$,
\begin{equation}\label{prot_filter}
\mathbf{h}=[h[0], \cdots ,h[L_{h}-1]]^{T}.
\end{equation}
The transfer function of the prototype
filter becomes
\begin{equation}\label{eq:resp_prot}
H(z)=\sum_{l=0}^{L_{h}-1}h[l]z^{-l}={\bf h}^{T}\phi(z)
\end{equation}
where $\phi(z)=[1,\cdots,z^{-(L_{h}-1)}]^{T}$.
The transformation between fullband signals to subband signals
depends prototype filter frequency response $H(e^{j\omega})$. The
desired filter should have a lowpass characteristic with a
normalized cut-off frequency $\omega_{p}=\pi/M$. Since it is
desirable to have an approximately linear phase in the passband, the
desired complex frequency response is specified as
\begin{equation}
H_{d}(e^{j\omega})=e^{j\omega\tau}, \hspace{1cm} \forall \omega
\in [-\omega_{p},\omega_{p}]
\end{equation}
where $\tau=(L_{h}-1)/2$.
The most common design criteria in filter design are least square
and min-max \cite{parks}. Since it is important to have a flat
passband response for the prototype filter, a least square
criterion is chosen in the cost function. However, a min-max
criterion is also included in the constraints to control and
ensure a low level of aliasing.
The least square and the min-max errors are given as
\begin{equation}\label{error1}
e_1({\bf h})=\frac{1}{2\omega_{p}}\int^{\omega_{p}}_{-\omega_{p}}|
H(e^{j\omega})-H_{d}(e^{j\omega})|^2d\omega
\end{equation}
and
\begin{equation}\label{minmax}
e_2({\bf h})=\max_{-\omega_{p}\leq \omega \leq \omega_{p} }
|H(e^{j\omega})-H_{d}(e^{j\omega})|,
\end{equation}
respectively.
It follows from (\ref{eq:resp_prot}) that $H(e^{j\omega})$ is a
linear function of the coefficient ${\bf h}$. Thus, (\ref{error1})
is reduced to a quadratic function of ${\bf h}$
\begin{eqnarray*}
e_{1}({\bf h})={\bf h}^{T} {\bf A} {\bf h}-2{\bf h}^{T}{\bf b}+1,
\end{eqnarray*}
where $\bf{A}$ is a $L_{h} \times L_{h} $ matrix
\begin{equation}
\mathbf{A}=\frac{1}{2\omega_{p}}\int^{\omega_{p}}_{-\omega_{p}}
\mathbf{\phi}(e^{j\omega})\mathbf{\phi}^{H}(e^{j\omega})d\omega,
\label{eq:C}
\end{equation}
and $\mathbf{b}$ is a $L_{h} \times 1$ vector
\begin{equation}
\mathbf{b}=\frac{1}{2\omega_{p}}\int^{\omega_{p}}_{-\omega_{p}}
{\cal R} \{e^{j\omega\tau}\mathbf{\phi}^{H}(e^{j\omega})\}d\omega.
\label{eq:b}
\end{equation}
The operators $[.]^{H}$ and ${\cal R}\{.\}$ denotes the Hermitian
transpose of a matrix and the real part of a complex number,
respectively.
The prototype filter is also designed to have a minimum in-band
aliasing effect. Since the transfer functions of the subband
filters are given by (\ref{eq:umf}), it is sufficient to
minimize the energy in the aliasing terms of the first subband.
The aliasing error can therefore be formulated as a quadratic function of
${\bf h}$ \cite{haan},
\begin{equation}\label{alias_cost}
\beta({\bf h})={\bf h}^{T}{\bf C}{\bf h},
\end{equation}
where $\mathbf{C}$ is a $L_{h} \times L_{h}$ matrix,
\begin{eqnarray*}
\mathbf{C}=\frac{1}{2\pi D}\sum^{D-1}_{d=1}\int^{\pi}_{-\pi}
\mathbf{\phi}(e^{j\omega/D}W^{d}_{D})\mathbf{\phi}^{H}(e^{j\omega/D}W^{d}_{D})d\omega.
\end{eqnarray*}
The resulting problem is given by a combination of the least square error
$e_{1}({\bf h})$ and the aliasing effect $\beta({\bf h })$ with
respect to some acceptable attenuation in the passband. A joint
cost function can be formulated as
\begin{equation}\label{eq:epsilon}
e_{1}({\bf h})+\beta({\bf
h})=\mathbf{h}^{T}(\mathbf{A}+\mathbf{C})\mathbf{h}-2\mathbf{h}^{T}\mathbf{b}+1.
\end{equation}
By combining the cost function (\ref{eq:epsilon}) with the min-max
constraints (\ref{minmax}), the following problem is obtained
\begin{equation}\label{eq:opt_prob}
\left \{\begin{array}{ll} \min\limits_{\bf h}
\mathbf{h}^{T}(\mathbf{A}+\mathbf{C})\mathbf{h}-2\mathbf{h}^{T}\mathbf{b}+1
\\ |H(e^{j\omega})-H_{d}(e^{j\omega})| \leq \epsilon, \hspace{1cm}
\forall \omega \in- [\omega_{p},\omega_{p}]
\end{array} \right.
\end{equation}
where $\epsilon$ is the maximum deviation in the passband. It
follows from the real rotation theorem \cite{parks} that a
magnitude inequality $|y| \leq \epsilon$ in the complex domain can
equivalently be expressed as
\begin{equation}\label{rr_theor}
{\Re}\{ ye^{j2\pi \lambda} \} \leq \epsilon, \hspace{.5cm}
\forall \lambda \in [0,1],
\end{equation}
where $y$ denotes a complex number. Using (\ref{eq:resp_prot}) and
(\ref{rr_theor}), (\ref{eq:opt_prob}) gives the final problem
\begin{equation}\label{semi_inf_pr}
\left \{\begin{array}{ll} \min\limits_{\bf h}
\mathbf{h}^{T}(\mathbf{A}+\mathbf{C})\mathbf{h}-2\mathbf{h}^{T}\mathbf{b}+1
\\ \Re\left\{ \left({\bf h}^{T}\phi(e^{j\omega})-H_{d}(e^{j\omega})\right)e^{j2\pi \lambda} \right\}
\leq \epsilon, \hspace{.2cm} \forall \omega \in-
[\omega_{p},\omega_{p}], \hspace{.2cm} \lambda \in [0,1].
\end{array} \right.
\end{equation}
The problem (\ref{semi_inf_pr}) has an finite number of variables
and infinite number of constraints. It is called a semi--infinite
quadratic programming problem. This problem can be solved by using:
(i) semi--infinite quadratic program technique or (ii)
conventional quadratic programming using discretization. In this
paper, we use the second method to design the prototype filter.
The problem (\ref{semi_inf_pr}) is then reduced to a quadratic
programming problem, which is solved by using standard
quadratic programming techniques.
\section{Closed--loop Subband CMA}
In Figure \ref{fig:cma}, the basic architecture of an adaptive CMA
equalizer is outlined. The output signal of the CMA equalizer $y(n)$
is given by
\begin{equation}
y(n)={\bf g}^{H}(n) {\bf x}(n),
\end{equation}
where $(\cdot)^H$ is the complex conjugate transpose. The input vector to
the equalizer is denoted as
$${\bf x}(n)= [x(n), x(n-1), \cdots, x(n-L+1)]^{T}.$$
\begin{figure}[!!!!h]
\centerline{\includegraphics[scale=.7]{cmafig.eps}}
\caption{An adaptive CMA equalizer.}
\label{fig:cma}
\end{figure}
By assuming that the transmitted signal, on average, has a constant signal
constellation, the equalizer should try to recover this
property. However, multipath fading causes amplitude fluctuations
in the received signal. Further, the received signal is
corrupted by additive white Gaussian noise.
The objective of CMA is to restore the output signal $y(n)$
to a constant signal constellation.
This is accomplished by adjusting the weight vector ${\bf
g}(n)$ in order to minimize the cost function $J$, which provides
a measure of the amplitude fluctuations. The cost function $J$ is
given by
\begin{equation}
J_{p,q}= E \bigg[ \Big( \vert y(n) \vert ^{p}-1 \Big)^q \bigg]
\end{equation}
The cost function $J$
can be minimized by using a gradient search algorithm. The
parameters $p$ and $q$ control the sensitivity and the convergence
behavior of the CMA. Smaller values of $p$ and $q$ usually
provide higher noise tolerance and stability, but slower convergence.
Typically, to ensure convergence and reasonably small tap-gain
fluctuation, $(p,q)$ are set to (1,2),(2,2),(1,1) or (2,1). By replacing the
ensemble average operation with an instantaneous value in the cost
function $J$, the update equation for ${\bf g}(n)$ is obtained as
\begin{eqnarray}\label{eq:cma}
{\bf g}(n+1) & = & {\bf g}(n) - \mu \varepsilon_{p}(n)^{*}
{\bf x}(n)\\
\varepsilon_{1,2}(n) &=& y(n) - \frac{y(n)}{\vert y(n) \vert}\label{eq:case1}\\
\varepsilon_{2,2}(n) &=& y(n)\Big(|y(n)|^{2}-1\Big)\\
\varepsilon_{1,1}(n) &=& \frac{y(n)}{|y(n)|} \mbox{sgn}\Big(|y(n)|-1\big)\\
\varepsilon_{2,1}(n) &=&
y(n)\mbox{sgn}\Big(|y(n)|^{2}-1\Big),\label{eq:case4}
\end{eqnarray}
where $\mu$ is the step-size constant and
$\varepsilon_{p,q}(n)$ is the Godard error signal representing
the cost function $J_{p,q}$.
It is clear from the definition of $J$ and the CMA
weight update equation that the algorithm is insensitive to
phase fluctuations. Conventional non-blind algorithms, such as
least mean squares (LMS) employ the mean square error (MSE) cost function to compensate
both the amplitude and phase of the signal by the use of a pilot signal.
However, due to the phase insensitivity of CMA, the carrier phase must
be recovered after the adaptive processing. The phase recovering
algorithm proposed by Godard \cite{godard} is a first order
\textsl{phase locked loop} (PLL).
The main drawback of the CMA is the relatively slow
convergence rate. This becomes increasingly significant as
wireless applications involving rapid changes in channel
characteristics become more common. In order to improve the speed
of convergence of the CMA the following
normalization factor has been used \cite{jones}
\begin{equation}
\eta(n)={\bf x}^{H}(n){\bf x}(n).
\end{equation}
The normalized CMA (NCMA) weight update equation is now given by
\begin{equation}
{\bf g}(n+1) = {\bf g}(n) - \frac{\mu\varepsilon_{p,q}(n)^{*}\mathbf{x}(n)}{\eta(n)}.
\label{eq:ncma}
\end{equation}
In the subband case the error signal
is created from the fullband signal and feeds back into the
subbands, thus giving a local subband weight update given as
\begin{equation}
{\bf g}_{m}(n+1) = {\bf g}_{m}(n) - \frac{\mu \varepsilon_{m}(n)^{*}{\bf x}_{m}(n)}{\eta_{m}(n)},
\label{eq:sncma}
\end{equation}
where $m, 0 \leq m \leq M-1$ is the subband index, $\varepsilon_{m}(n)$
denotes the $m^{th}$ subband component of $\varepsilon_{p,q}(n)$,
$\eta_{m}(n)={\bf x}_{m}^{H}(n){\bf x}_{m}(n)$ denotes
the normalization factor in the $m^{th}$ subband and
$${\bf x}_{m}(n)=[x_{m}(n), x_{m}(n-1), \cdots, x_{m}(n-L_{s}+1)]^{T}$$
is the $m^{th}$ subband component of ${\bf x}(n)$, where $L_{s}$ is
the length of the subband equalizer. The closed loop subband NCMA
is thus given by a combination of a fullband error calculation, a
local subband weight update and a transformation of the subband
weights into a corresponding fullband filter.
Since, it is desirable to have the main energy of the equalization
filter close to the center of the filter, the mid tap of both the
subband- and fullband- equalization filters are initialized to $1$.
\section{Subband--to--Fullband Transformation Methods}\label{stack}
In this section, two different transformation methods are proposed
for obtaining the fullband filter weights from subband weights.
The methods are generic and can be used for any subband to
fullband weight transformation.
\subsection{DFT-1 Stacking Method}
The original transformation or stacking method (DFT-1) was suggested
in \cite{MThi95} for obtaining the fullband response from the
subband responses, in case of a real channel.
These results have been extended to include complex valued channels.
The complex channel allows amplitude
attenuation and phase shift for the transmitted signals, as in the case of
wireless communication.
For $M$ subbands with a decimation factor of $D=\frac{M}{2}$, the
length of each subband equalizer is $L_{s}$ where $L_{s}=2L/M$.
Denote
\begin{equation}
{\bf g}_{m}=[g_{m}(0),\cdots,g_{m}(L_{s}-1)], \hspace{.2cm} 0 \leq m \leq
M-1
\end{equation}
as the equalizer coefficients for the $m^{th}$ subband. The
corresponding $L_{s}$ point discrete Fourier transform (DFT) of
${\bf g}_{m}$ is given as
\begin{equation}
{\bf G}_{m}=\left[G_{m}(0),\cdots,G_{m}(L_{s}-1)\right].
\end{equation}
The fullband frequency response
\begin{equation}{\bf G}=[G(0),\cdots,G(L-1)]
\end{equation} is obtained by stacking the
frequency of the subband responses by using the following procedure.
\begin{procedure}\label{proc_1} DFT-1 stacking method
\begin{itemize}
\item Step 0: For the $0^{th}$ subband we have
\begin{equation}
G(l)=G_{0}(l), \hspace{.2cm} 0 \leq l \leq L_{s}/4-1.
\end{equation}
\item Step 1: For $m=1$ to $M-1$, the frequency response ${\bf G}_{m}$ is stacked
to ${\bf G}$ as follows:
\begin{itemize}
\item If $m$ is odd, then
\begin{equation}
G \left(l+(m-1)L_{s}/2\right)=G_{m}(l)
\end{equation}
for $~ L_{s}/4\leq l \leq 3L_{s}/4-1$.
\item If $m$ is even, then
\begin{equation}
G(l+(m-2)L_{s}/2)=G_{m}(l)
\end{equation}
for $~ 3L_{s}/4 \leq l \leq L_{s}-1$
and
\begin{equation}
G(l+mL_{s}/2)=G_{m}(l)
\end{equation}
for $~ 0\leq l \leq L_{s}/4-1$.
\end{itemize}
\item Step 2 : The last $L_{s}/4$ points of ${\bf G}$ are obtained by stacking the $0^{th}$
subband
\begin{eqnarray*}
G(l+(M-2)L_{s}/2)=G_{0}(l)
\end{eqnarray*}
for $~ 3L_{s}/4 \leq l \leq L_{s}-1$.
%
\end{itemize}
\end{procedure}
The fullband equalizer ${\bf g}$ is the inverse DFT of ${\bf G}$.
\subsection{DFT-2 Stacking Method}
The DFT-2 stacking approach \cite{Huo} is modified for a complex
channel. This approach is implemented by calculating a $2L_{s}$ point DFT based on the
$L_{s}$ tap filter in each subband. By doubling the number of points for DFT,
the performance of subband equalizer is improved since finite block effects
can be avoided.
The DFT of $[g_{m},{\bf 0}]$ is denoted by
\begin{equation}
{\bf G}'_{m}=[G'_{m}(0),\cdots,G'_{m}(2L_{s}-1)],
\end{equation}
where ${\bf 0}$ is a $1 \times L_{s}$
vector of 0. The frequency response for the fullband filter
\begin{equation}
{\bf G}'=[G'(0),\cdots,G'(2L-1)]
\end{equation}
is formed from the subband frequency responses ${\bf G}'_{m}$ by modifying
Procedure \ref{proc_1}.
\begin{procedure}\label{proc_2}\textit{DFT-2 stacking method}
\begin{itemize}
\item Step 0: For the $0^{th}$ subband we have
\begin{equation}
G'(l)=G'_{0}(l), \hspace{.2cm} 0 \leq l \leq L_{s}/2-1.
\end{equation}
\item Step 1: For $m=1$ to $M-1$, the frequency response of the $m^{th}$ subband
${\bf G}'_{m}$ is stacked to ${\bf G}'$ as follows:
\begin{itemize}
\item If $m$ is odd, then
\begin{equation}
G' \left(l+(m-1)L_{s}\right)=G'_{m}(l)
\end{equation}
for $~ L_{s}/2\leq l \leq 3L_{s}/2-1$.
\item If $m$ is even, then
\begin{equation}
G'(l+(m-2)L_{s})=G'_{m}(l)
\end{equation}
for $~3L_{s}/2 \leq l \leq 2L_{s}-1$ and
\begin{equation}
G'(l+mL_{s})=G'_{m}(l)
\end{equation}
for $~ 0\leq l \leq L_{s}/2-1$.
\end{itemize}
\item Step 2: The last $L_{s}/2$ points of ${\bf G}'$ are
obtained by stacking the $0^{th}$ subband
\begin{equation}
G'(l+(M-2)L_{s})=G'_{0}(l)
\end{equation}
for $~ 3L_{s}/2 \leq l \leq 2L_{s}-1$.
\end{itemize}
\end{procedure}
The impulse response for the fullband equalizer is obtained by
truncating the last $L$ points of a $2L$ point inverse DFT of
${\bf G}'$.
\chapter{Simulation Results}\label{ch:res}
The Hiperlan II inspired channel model is used to evaluate and
compare the performance of a closed--loop delayless subband
equalizer with a fullband equalizer. The filter bank design method
presented in this paper, is bench marked against a standard design
filter bank. This standard filter bank is denoted FIR\_1, since the
prototype filter is designed by using Matlab \emph{fir1} function
\cite{matlab}. The \emph{fir1} function is based on the windowing
method. A Hanning window is used with the cut-off-frequency
$\frac{1}{2M}$. For both filter bank designs, the length of the
prototype filter is chosen to be four times the total number of
subbands and the weighting factor in (\ref{eq:opt_prob}) is one.
Note, that if nothing else is mentioned the filter bank is
designed by the optimization formulation given in Equation
(\ref{semi_inf_pr}).
To evaluate the convergence properties of the subband equalizer,
the average fullband amplitude fluctuation $\varepsilon_{A}(m)$
from the constant constellation $R_{s}$ is being used, since
several error functions are compared against each other. The
amplitude fluctuation measure is defined as
\begin{equation}
\varepsilon_{A}(m) =10 \log_{10}\left(
\frac{1}{N_{B}}\sum\limits_{n=0}^{N_{B}-1}|
|u(n)|-R_{s}|^{2}\right), \label{eq:ea}
\end{equation}
where $N_{B}$ is the block length over which the amplitude
fluctuations are averaged. The block length is set to 1000
symbols. In all convergence simulations, if nothing else is being
stated, the signal to noise ratio (SNR) is 30 dB. The steady state
performance is evaluated by using the BER measure for varying SNR.
It is denoted as "a steady state" comparison, since it is assumed
that the algorithm has converged when the actual BER is being
calculated.
In Figure \ref{fig:res1}, the average fullband amplitude
fluctuation $\varepsilon_{A}(m)$ has been used to compare four
different versions of CMA. The different algorithms use the four
error functions $ \varepsilon_{1,1}$, $\varepsilon_{2,1}$,
$\varepsilon_{1,2}$ and $\varepsilon_{2,2}$, which are given
by (\ref{eq:case1}) - (\ref{eq:case4}).
They all use $M= 64$ subband and a fullband FIR length of 256
taps ($L=256$). The two error functions derived from a power measure $q=2$
clearly outperform the other two, at least when using the measure defined
in (\ref{eq:ea}). The simulation results also show that the
error function $\varepsilon_{2,2}$ results in a much faster
algorithm than the $\varepsilon_{1,2}$ one. For our purposes, the
$\varepsilon_{2,2}$ will be used in the following.
In Figure \ref{fig:res2}, the fullband equalizer (M = 1) and
subband equalizers are compared for a varying number of subbands.
Both equalizers are using $L=256$ taps, and it is clear from
the results shown in Figure \ref{fig:res2} that the major gain in
using a subband uppdate approach lies in the improved convergence
rate. The 64 subband equalizer is roughly 12 times faster
than the fullband equalizer.
The subband equalizer performance, measured by using the average
fullband amplitude fluctuation $\varepsilon_{A}(m)$ for different
combinations of filter lengths $L$ and number of subband $M$,
is presented in Figure \ref{fig:res3}. The choice of filter
lengths and the number of subbands are chosen so that the number of
coefficients $L_{s}=L/D$ in each subband is kept the same.
Finally, the subband results have been compared with the
performance of a fullband $L=256$ tap equalizer. The
convergence for the different settings of the subband equalizers
are more or less the same while there are large differences in
steady state performance, especially when comparing the $L=32$ tap
and the $L=256$ tap equalizer.Further simulations and evaluations
will be based on the $M=64$ subband equalizer with $L=256$ taps.
To extend the equalizer beyond 256 taps will only result in a marginal
improvement in the performance.
Figures \ref{fig:res4} - \ref{fig:res6} show the importance of
using a proper subband--to--fullband transformation method as well as
a proper designed filter bank. The SNR has been set to 30 dB. It is
notable that by using the DFT-2 transformation method it is almost
possible to compensate for the poor alias suppression of the
FIR\_1 filter bank, see Figure \ref{fig:res6}. To clarify the
performance difference between using the DFT-1 method and the DFT-2
method, when the optimized filter bank is being used, the comparison
is re-made in a noise free setting, see Figure \ref{fig:res7}. The results show
that the DFT-2 outperforms the DFT-1 method.
In Figure \ref{fig:res8}, the steady state solution of the
fullband equalizer is compared to the 64 subband equalizer. The
evaluation compares the BER. This evaluation
shows that there is a very good match between subband
and fullband solutions.
\begin{figure}[h]
\centerline{\includegraphics[scale=.7]{conv_256_64.eps}}
\caption{The fullband equalizer performance using the four different
error functions with an $L=256$ taps equalizer and SNR = 30 dB.}
\label{fig:res1}
\end{figure}
\begin{figure}[h]
\centerline{\includegraphics[scale=.7]{conv_256_e22.eps}}
\caption{The performance of the subband equalizer for different
number of subbands, with an $L=256$ taps equalizer and SNR = 30
dB.} \label{fig:res2}
\end{figure}
\begin{figure}[h]
\centerline{\includegraphics[scale=.7]{conv_e22.eps}} \caption{A
comparison between fullband and subband equalization by using
different combinations of equalizer lengths and number of
subbands. SNR = 30 dB.} \label{fig:res3}
\end{figure}
\begin{figure}[h]
\centerline{\includegraphics[scale=.7]{fb_comp_256_64.eps}}
\caption{A comparison between the two filter bank designs for an
$M=64$ subband equalizer, $L=256$ and SNR = 30 dB, measured by
$\varepsilon_{A}(m)$.} \label{fig:res4}
\end{figure}
\begin{figure}[h]
\centerline{\includegraphics[scale=.7]{dft_comp_256_64.eps}}
\caption{A comparison of the two different subband--to--fullband
transformation methods when $M=64$ and the \emph{fir1} filter
bank is employed. The SNR is 30 dB.} \label{fig:res5}
\end{figure}
\begin{figure}[h]
\centerline{\includegraphics[scale=.7]{dft2_256_64.eps}}
\caption{A comparison of the two filter bank designing methods when a 64
subband equalizer uses the DFT-2 subband--to--fullband
transformation method. The SNR is 30 dB. }
\label{fig:res6}
\end{figure}
\begin{figure}[h]
\centerline{\includegraphics[scale=.7]{fbdft_256_64.eps}}
\caption{A convergence comparison of the two different
subband--to--fullband transformation methods, when a 64 subband equalizer
uses the optimized filter bank in a noise free environment.}
\label{fig:res7}
\end{figure}
\begin{figure}[h]
\centerline{\includegraphics[scale=.7]{berres.eps}}
\caption{BER comparison between a fullband equalizer and a subband equalizer employing 64 subband.}
\label{fig:res8}
\end{figure}
\chapter{Conclusions}\label{ch:con}
In this paper, a new delayless subband closed--loop CMA equalizer
is proposed. A filter bank design method has been presented that
minimize the aliasing in the subband signals. By using this filter
bank method a dramatic performance improvement of the subband
equalizer has been obtained.
Further, two different subband to fullband transformation methods have
been presented and comparedr. Simulations have been
carried out for a 100 taps sparse complex channel. Moreover, whereas the
convergence rate is significantly improved as the number of
subbands increases, while the cost is only a very small degradation
in the bit error rate. An $L = 256$ tap equalizer using $M=64$
subbands, has 12 times faster convergence rate compared to its
fullband counterpart. It has also been shown that the DFT-2 method
outperforms the DFT-1 method. The DFT-2method clearly avoids the block
effects, and will result in a fullband filter that better
represents the fullband processing. The DFT-2 method also
compensates, to some degree, for a non-optimal choice of filter bank.
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